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6. Production Definition 6.1. Production Is the Transformation of a Vector Definition 6.3. a Production Set Y

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6. Production Definition 6.1. Production Is the Transformation of a Vector Definition 6.3. a Production Set Y EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 251 6. Production Definition 6.1. Production is the transformation of a vector of commodities into a different vector of commodities. Production can be organized in either of two environments: • In hierarchical environments: within firms In market environments: between firms Important questions without clear answers: Why do firms exist? ◦ What determines their boundaries? ◦ Economists understand more about the market environment than • about the hierarchical environment. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 252 We use a simple model of production within the firm in order to • generate supply curves. Supply curves are analyzed within the market environment. • Definition 6.2. A production activity or y is a vector of inputs (negative) and outputs (positive) that describes a transformation of some goods into others. Definition 6.3. A production set Y Rn is the set of all activities that are physically possible to⊂ implement. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 253 Suppose the commodities are given by: • ice cream 2 cheese 1 ⎡ ⎤ ,andsupposey = ⎡ ⎤ Y. milk 6 ∈ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − ⎥ ⎢ sugar ⎥ ⎢ 8 ⎥ ⎢ ⎥ ⎢ − ⎥ ⎣ ⎦ ⎣ ⎦ Then it is physically possible to convert 6 units of milk and 8 units • of sugar into 2 units of ice cream and 1 unit of cheese. Real production in modern economies is extraordinarily • complicated. Vectors of inputs and outputs would be very large. • EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 254 Think about the economics department at BU: Inputs: • Professors, each one with different skills, training. Staff, again, each with different skills, training. Structure (this building) is specialized. Equipment, not too important. Outputs: • Undergraduate Masters’ training Doctors’ Training Research of doctoral students Research of professors EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 255 How could you represent in a vector what we do in the economics • department? How could you represent all the possible production activities in • the department, including those that have not been used here? EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 256 The economists’ model is extremely abstract! • The set Y , below, is the production set. • Y does not contain any points in Rn except 0. Why not? • + Think of the an activity vector y Y as an arrow from the origin • that indicates a decrease in the stocks∈ of inputs and an increase in the stocks of outputs. n y . + Y 0 n . − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 257 If a vector of initial resources, z0 , is added to each point in the • set Y ,aproduction possibility set (PPS) Y + z = y + z y Y is obtained. 0 { 0 | ∈ } In the diagram below, the activity y transforms the initial • resources z0 to the the vector of resources z1 . z + Y 0 z1 y Production z0 Possibility Set 0 EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 258 The PPS lies in Rn . • + Contains all the combinations of commodities that can be • obtained in the economy. z corresponds to 0 in the original production set. • 0 z0 is the vector obtained if no production occurs. • EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 259 ThefollowingarecommonlyassumedtobepropertiesofY : Assumption 6.1. Continuity in production: Y is closed. Assumption 6.2. Inactivity is possible, but there are no free products: Y R+ = 0 . ∩ { } You can’t get outputs without using inputs. • But you can do nothing, so 0 Y • ∈ z n . + {y≤z} Y P(x) 0 x n . − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 260 For any x Y , we define the higher-productivity set, P(x), by • P(x)= y∈ y Y , y x . { | ∈ ≥ } As compared with x,allthepointsinP(x) (except for x itself) either use smaller amounts of some inputs ◦ and/or yield greater amounts of some outputs ◦ z n . + {y≤z} Y P(x) 0 x n . − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 261 Assumption 6.3. Limited productivity: Given any x Y , the higher-productivity set P(x) is • bounded. [Missing∈ from M-C, but should be there.] Assumption 6.4. Free disposability: If z Y ,andy z ,theny Y . • ∈ ≤ ∈ It is always possible to waste inputs. • See graph. • Assumption 6.5. Irreversibility: if y Y ,then y / Y . • ∈ − ∈ Can’t turn process around, produce inputs from outputs. Technical assumption: Labor, energy used in production, cannot be fully recovered. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 262 Assumption 6.6. Convexity: Y is a convex set. Strong assumption. • Rules out economies of scale. • Graph below violates convexity, shows increasing returns. • n . + Y 0 n . − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 263 Assumption 6.7. Additivity: If x, y Y ,thenx + y Y . • ∈ ∈ Strong assumption. Rules out diseconomies of scale. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 264 Assumption 6.8. Y is a convex cone: If x, y Y ,thenfor α, β > 0, αx + βy Y . ∈ ∈ Very strong assumption. • Implies additivity. • Mathematically, Y is closed • under linear combinations. Implies constant returns to scale. • Y 0 EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 265 Replication argument: • Production sets have additivity property between two different production activities if they don’t interfere with one another. Production activities describe possible transformations, not resource constraints. If a production activity exits, then it ought to be possible to replicate (duplicate) it exactly, as many time as desired. = If y Y , then for any integer n, ny⇒ Y .∈ ∈ = Y has additivity. ⇒ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 266 Proposition 6.1. If 0 Y ,andY is convex and has additivity, then Y is a convex∈ cone. Proof. We have: By convexity, if y Y , then for α [0, 1], • αy = αy +(1 α∈)0 Y . ∈ − ∈ For α > 1, find integer n α. • ≥ By additivity, ny Y ∈ By convexity αny Y . n ∈ = αy Y . ⇒ ∈ If x, y Y , then for α, β > 0, αx, βy Y . • ∈ ∈ By additivity: αx + βy Y . • ∈ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 267 Suppose Y Rn is convex but not a convex cone. • ⊂ By assuming that there is a “hidden input,” z , we can create Y 0 • that is a convex cone. Assume that each y Y , uses also one unit of z . • ∈ y For each y Y , set y 0 = . • ∈ 1 " − # y n+1 Define Y 0 = α α 0, y Y R . • ( " 1 # ¯ ≥ ∈ ) ⊂ − ¯ ¯ Y is a convex cone. ¯ • 0 ¯ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 268 Definition 6.4. If there is only one output, q, and a number of inputs, denoted by the vector z ,thenifY is a closed set, has free disposability with 0 Y ,wedefine a production function f by ∈ q f(z)=max q Y ( ¯ " z # ∈ ) ¯ − ¯ ¯ ¯ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 269 Definition 6.5. AproductionsetY exhibits increasing returns to scale if for any y Y and α > 1, there is a point y > ay with y Y . ∈ 0 0 ∈ If you increase the scale of production, then it is possible to • increase productivity. Definition 6.6. AproductionsetY exhibits decreasing returns to scale if for any y Y and α (0, 1), there is a point y > ay with y Y . ∈ ∈ 0 0 ∈ If you decrease the scale of production, then it is possible to • increase productivity. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 270 Definition 6.7. AproductionsetY exhibits nonincreasing returns to scale if for any y Y and α (0, 1), ay Y . ∈ ∈ ∈ It is always possible to decrease the scale of production. • Definition 6.8. AproductionsetY exhibits nondecreasing returns to scale if for any y Y and α > 1, ay Y . ∈ ∈ It is always possible to increase the scale of production. • Definition 6.9. AproductionsetY exhibits constant returns to scale if for any y Y and α > 0, ay Y . ∈ ∈ Nonincreasing and nondecreasing returns to scale. • EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 271 Proposition 6.2. Let f (x) with x Rn be a concave ∈ + production function. Define a hidden input z R+ and a production function ∈ 1 F (X, z) zf X . ≡ z µ ¶ Then F is concave, has constant returns to scale, and F (x, 1) f (x). ≡ The quantity • 1 x X ≡ z should be interpreted as the quantity of X per unit of the hidden input. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 272 If you think of the hidden input z as management, this says that • each separate unit of managment can operate the production process described by f (x). Therefore, a total of z units of management and X = zx units of • the original input gives us zf (x) units of output. This suggests that if a production function has decreasing returns • to scale, it is because of a hidden input that is not increased as the scale of production increases. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 273 Once the hidden input is introduced, the new production function • is concave and has constant returns to scale. Equivalently, the set under the graph of the new production • function is a convex cone. FXz(), fx() z 1 x, X 0 EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 274 FXz(), fx() z 1 x, X 0 EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 275 Proof. We have 1 F (αX, αz)=αzf αX = αF (X, z) , αz µ ¶ which proves that F has constant returns to scale.
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